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![Terminal velocity of air bubbles in water at 20 • C with different level of contamination as a function of the diameter according to various sources. Picture reprinted from Clift et al. [8] with permission.](profile/Tobias-Kempe/publication/277577221/figure/fig2/AS:667786376380417@1536224087280/Terminal-velocity-of-air-bubbles-in-water-at-20-C-with-different-level-of-contamination.png)
Terminal velocity of air bubbles in water at 20 • C with different level of contamination as a function of the diameter according to various sources. Picture reprinted from Clift et al. [8] with permission.
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The numerical simulation of spherical and ellipsoidal bubbles in purified fluids requires the imposition of the free-slip boundary condition at the bubble surface. This paper describes a numerical method for the implementation of free-slip boundary conditions in the context of immersed boundary methods. In contrast to other numerical approaches for...
Contexts in source publication
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... survey of experimental data for terminal velocities of air bubbles in pure and in contaminated water from [8] is presented in Figure 1 with logarithmic axes. Surface-active contaminants affect the rise velocity most strongly in the upper spherical and in the ellipsoidal regime. ...
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... shown in Figure 1, the difference in the rise velocity of the bubbles with and without contamination successively increases starting from the lowest Reynolds number Re up to the peak value at Re ≈ 450 and then decreases again in the regime of unsteady separation and further in the regime of turbulent flow. But also for these higher Reynolds numbers the bubble motion is significantly affected by the boundary condition at the liquid-gas interface. ...
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... third variant is an extension of the latter system, specified by the additional requirement that the tangential directions coincide with the direction of principal curvature. It is called the local coordinate system, S l , and is described in more detail with the help of Figure 13 in Section 5 below. In contrast to S c , the coordinate lines are not straight, but curved according to the radius of curvature in the respective direction. ...
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... an IBM is employed with a relatively modest number of grid points for a given curvature of the interface, this correction is essential as illustrated by the following test. The numerical results obtained when applying the improved method (40) to the test case defined in Section 2.3 above are displayed in Figure 10. The distance of the test point from the interface was chosen to be d M = 2 h. ...
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... this section the issue is investigated for spherical particles at Re = 1 and two different resolutions of the sphere (higher Re will be addressed below). displayed in Figure 11, and the results on the fine grid are shown in Figure 12. Obviously, no significant impact of the distance of the test point on the flow is observed for such very low values of Re. ...
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... this section the issue is investigated for spherical particles at Re = 1 and two different resolutions of the sphere (higher Re will be addressed below). displayed in Figure 11, and the results on the fine grid are shown in Figure 12. Obviously, no significant impact of the distance of the test point on the flow is observed for such very low values of Re. ...
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... case of a spherical particle obviously corresponds to the special case where R 1 = R 2 = R with R the radius of the sphere. As mentioned in Section 4.1 above, a local coordinate system S l is introduced at each Lagrange marker point coinciding with the interface in the vicinity of this point ( Figure 13). The main difference to the system in Section 4.5 above is that this coordinate system matches the surface only in the vicinity of this specific marker point and not necessarily in the vicinity of other markers. ...
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... latter are given by the corresponding angles to the normal direction. An illustration is provided in Figure 13. With these conventions the local coordinate system is defined and the shear stresses now need to be determined in this system to impose free-slip conditions. ...
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... the previous sections have dealt with fixed particles, we now consider objects with prescribed rotational as used for Case E2 above. A schematic of the configuration is shown in Figure 18. The angular velocity was set to ω x = U ∞ / R = 8.333 for both cases so that the maximum velocity of the interface in circumferential direction is equal the the free-stream velocity U ∞ . ...
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... results for the velocity in streamwise direction are displayed in Figure 19a, and in 19b the velocity in the direction transverse to the mean flow is shown. As expected, the fluid is carried along by the moving surface of particle in the no-slip case. ...
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... convergence of the grid was checked again to obtain mesh independent data. The results obtained with the IBM on a grid with N = 128 grid points are presented in Figure 21. With the pseudo free-slip condition the streamwise velocity in the wake is The dependency of the numerical results for no-slip and free-slip interfaces on the spatial resolution of the domain can also be addressed. ...
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Citations
... The study of free-slip boundary condition (BC) is crucial at certain occasions in order to imitate or to access its impact on the whole fluid flow pattern especially at a very low pressure [2] or when immersion or penetration of bubble or fluid particle into solid interface happens and results as surface contamination. Thus, the large viscosity ratio of the fluid near the solid boundary happens and causes in the rise of tangential velocity near the solid boundary [3]. In some cases, no-slip BC is well-posed in modeling of viscous flow without penetration condition but it is amiss when analysis of highly inviscid flows is considered. ...
Numerical simulations of an unsteady laminar lid-driven cavity Newtonian flow are executed to demonstrate formation of eddies under different aspect ratios and steady-state criteria. A couple of nonlinear unsteady partial differential equations (PDEs) of Navier Stokes satisfying a set of boundary conditions is examined with both inclusions of no-slip and free-slip effects. Finite difference method (FDM) is employed to solve the vorticity-stream function using a self-developed Matlab ® code embedded in a GUI to ease the cavity problem analysis by the end users. Four models are studied where Ansys Fluent ® employing finite element method is used to verify the present FDM steady-state results produced for the first model. It is observed that the value of stream function begins to drop and streamline distribution changes shape when e ≤ 10 ⁻⁵ . Refrainment of merging of secondary eddies also happens until the free-slip boundary condition effect passes the threshold value. Stretching effect with free-slip BC in the fourth model regulates the fluid dynamics to reach the entire cavity sufficiently with no room for eddy formation by increasing the slip length to an intense value. Free-slip simulates free surface applications in geophysical occurrences (river and glacier), spilling dynamics, ship hull designs, technologies of coating and fuel spraying/injection.
... Prescribing slip boundary conditions on curved boundaries is challenging and requires more complex machinery from theoretical and numerical viewpoints. Indeed, few works [43][44][45]49,50] have specifically addressed the curved ...
The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.
... Prescribing slip boundary conditions on curved boundaries is challenging and requires more complex machinery 100 from theoretical and numerical viewpoints. Indeed, few works [43,44, 45,49, 50] have specifically addressed the curved boundaries issue and provide a systematic numerical assessment of the accuracy and convergence properties of the proposed methods. It is worth mentioning that none of these works effectively achieves more than second-order of convergence, even with simple circular geometries. ...
The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three- dimensional fluid flow problems governed by the incompressible Navier-Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements, while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier-Stokes equations are then discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.
... Finally, the local forcing is transferred to surrounding Eularian locations using the regularized Dirac delta function. Similarly, this back-and-forth mechanism between Lagrangian and Eulerian locations to transfer the quantities is also used by Kempe & Fröhlich (2012); Kempe et al. (2015). In addition, Kempe et al. (2015) showed that even when the tangential force is set to zero, the standard direct forcing procedure leads to artificial shear stress at the fluid-solid body interface. ...
... Similarly, this back-and-forth mechanism between Lagrangian and Eulerian locations to transfer the quantities is also used by Kempe & Fröhlich (2012); Kempe et al. (2015). In addition, Kempe et al. (2015) showed that even when the tangential force is set to zero, the standard direct forcing procedure leads to artificial shear stress at the fluid-solid body interface. Because the fluid velocity and viscosity are considered the same inside and outside of the immersed boundary. ...
... Bubbles in purified liquids exhibiting a slip-boundary at their surface require a different forcing approach. This was developed by Kempe et al. [61], with particular account of the surface curvature, so that this case can be covered as well. ...
The article describes direct numerical simulations using an Euler–Lagrange approach with an immersed-boundary method to resolve the geometry and trajectory of particles moving in a flow. The presentation focuses on own work of the authors and discusses elements of physical and numerical modeling in some detail, together with three areas of application: microfluidic transport of spherical and nonspherical particles in curved ducts, flows with bubbles at different void fraction ranging from single bubbles to dense particle clusters, some also subjected to electro-magnetic forces, and bedload sediment transport with spherical and nonspherical particles. These applications with their specific requirements for numerical modeling illustrate the versatility of the approach and provide condensed information about main findings.
... The viscous normal stress on the wall is often negligible but can be significant on a free-slip surface. For example, for Stokes flow around a sphere having free-slip boundary condition on its surface, two-third of drag is caused by the viscous normal stress and one third is caused by the pressure [43]. ...
The instantaneous motion of a spherical particle in a channel flow is governed by the forces experienced by the particle. The magnitude and direction of the forces depend on the particle to channel size ratio, particle position, nature of the sphere surface (sticky/slippery), fluid properties and relative velocity between the fluid and the particle. In this work, we report the lift, the force component directed normal to the streamwise direction, on two classes of spheres, sticky and Janus, in a channel of square cross-section. The Janus spheres considered have both sticky and slippery hemispheres with the boundary between the two hemispheres parallel to the channel midplane. The effect of particle to channel size ratio, dimensionless particle position and particle Reynolds number on the lift are studied. The Janus sphere placed at the channel centerline is observed to experience the lift directed from the sticky to the slippery hemisphere. A correlation is proposed to predict the lift on the Janus sphere placed at the centerline of the channel. A sticky sphere positioned close to the channel wall experiences a significant lift directed away from it. For the Janus sphere placed at an off-center position two possibilities arise-slippery hemisphere facing the channel centerline (case A) or sticky hemisphere facing the channel centerline (case B). For case A, the lift is always directed away from the wall. For case B, the direction of lift depends on the particle position as well as particle Reynolds number. The moment coefficients for the sticky and Janus sphere are also presented.
... The full boundary integral implementation and single layer-63 only implementations differed in predicted values of rate of working by only 1.6-2.0%, a 64value comparable to the regularization and discretization errors. Increasing from a peak 65 radius of 0.02 body lengths to 0.06 body lengths, the progressive velocity was reduced by66 around 68% and mean rate of working increased by 72%, confirming the major propulsive67 advantage of slenderness for undulating swimmers. These results follow a similar trend to68 ...
... Stokes law, F = 6πaU where a is the radius of the sphere, and the solution to problem [F] is F = 4πaU[66]. Problem [P] has solution[63], ...
The method of regularized stokeslets is widely-used to model microscale biological propulsion. The method is usually implemented with only the single layer potential, with the double layer potential being neglected, despite this formulation often not being justified a priori due to non-rigid surface deformation. We describe a meshless approach enabling inclusion of the double layer which is applied to several Stokes flow problems in which neglect of the double layer is not strictly valid: the drag on a spherical droplet with partial slip boundary condition, swimming velocity and rate of working of a force-free spherical squirmer, and trajectory, swimmer-generated flow and rate of working of undulatory swimmers of varying slenderness. The resistance problem is solved accurately with modest discretization on a notebook computer with the inclusion of the double layer ranging from no-slip to free slip limits; neglect of the double layer potential results in up to 24% error, confirming the importance of the double layer in applications such as nanofluidics, in which partial slip may occur. The squirming swimmer problem is also solved for both velocity and rate of working to within a few percent error when the double layer potential is included, but the error in the rate of working is above 250% when the double layer is neglected. The undulating swimmer problem by contrast produces a very similar value of the velocity and rate of working for both slender and non-slender swimmers, whether or not the double layer is included, which may be due to the deformation’s `locally rigid body’ nature, providing empirical evidence that its neglect may be reasonable in many problems of interest. Inclusion of the double layer enables us to confirm robustly that slenderness provides major advantages in efficient motility despite minimal qualitative changes to the flow field and force distribution.
... For instance, Schwarz et al. [65] tried to utilize an immersed boundary method to observe an impact of flows on bubble shapes. Kempe et al. [66] imposed the free-slip boundary condition on a spheroidal particle fixated to center in a domain. With these methods or more advanced algorithm, it could be possible to investigate various features of a millimetric bubble in various conditions in future. ...
Path instability of a millimetric spheroidal bubble in quiescent fluid and in isotropic turbulence is investigated by direct numerical simulation. An immersed boundary method along with a new formulation of the equation of bubble motion is utilized to impose the no-slip condition on the surface of an air bubble in fixed shape with the equivalent diameter of 1.0∼4.0mm in contaminated water. The range of Galilei number defined as the ratio of the gravitational force to the kinematic viscosity considered in this study is 100 ∼ 800. In still fluid, as the bubble size grows, the frequency of the zigzagging motion of the bubble increases while the range in the orientation angle variation of the bubble is hardly affected. The effect of background turbulence on path instability of a rising bubble, which typically shows zigzag pattern in still fluid, is investigated at three different Reynolds numbers, Reλ, of 26, 45, and 73, or equivalently, for the ratio of fluid root-mean-square velocity to the bubble rise velocity u′/VT ranging 0.030 ∼ 0.671. When a bubble rises in isotropic turbulence, the terminal rise velocity of the bubble does not show a noticeable difference. However, the pathways are significantly distorted by turbulence. Furthermore, the magnitude of zigzagging frequency and the degree of obliquity of the bubble become enhanced with Reλ. We also observed wakes behind the bubble to find that the rear tails become weaker and tangled due to turbulence.
... The high flexibility of the explicit direct-forcing IBM is quite impressive. It has been successfully employed over the past two decades in a broad spectrum of fields, particularly in the simulation of particulate flows [30][31][32][33][34][35], thermally driven confined flows [36][37][38][39], and two-phase immiscible flows [40] and in the phenomenological modeling of the mobility and growth of cancerous tumors [41][42][43]. ...
An extended immersed boundary methodology utilizing a semi-implicit direct forcing approach was formulated for the simulation of incompressible flows in the presence of periodically moving immersed bodies. The methodology utilizes a Schur complement approach to enforce no-slip kinematic constraints for immersed surfaces. The methodology is split into an “embarrassingly” parallel pre-computing stage and a time integration stage, both of which take advantage of the general parallel file system (GPFS) for efficient writing and reading of large amounts of data. The methodology can be embedded straight forwardly into the whole family of pressure–velocity segregated solvers of incompressible Navier–Stokes equations based on projection or fractional step approaches. The methodology accurately meets the no-slip kinematic constraints on the surfaces of immersed oscillating bodies. In this study, it was extensively verified by applying it for the simulation of a number of representative flows developing in the presence of an oscillating sphere. The capabilities of the methodology for the simulation of incompressible flow generated by a number of bodies whose motion is governed by general periodic kinematics were demonstrated by simulation of the flow developing in the presence of two out-of-phase oscillating spheres. The physical characteristics of the generated flows in terms of the time evolutions of the total drag coefficients were presented as a function of Reynolds values. The vortical structures inherent in the generated flows were visualized by presenting the isosurfaces of theλ2 criterion.
... For momentum transfer at the boundary surface, the no-slip and no-penetration boundary conditions are usually enforced on solid surfaces. Those two boundary conditions indicate there are no relative velocities at the surface (Day 1990), mathematically u| Γ = 0 and v| Γ = 0, where u| Γ and v| Γ are relative velocity in the tangential and perpendicular directions at the interface Γ. Another extensively used boundary condition for momentum transfer is the slip boundary condition (Kempe et al. 2015). In the cases of rarefied-gas dynamics and non-Newtonian fluids in which the Newtonian law of viscosity cannot hold, modeling the solid surface as slip boundary would be more physically accurate (Lauga et al. 2007). ...
... Investigation of flows over such slip surface is quite important as non-Newtonian fluid is frequently encountered in chemical and process engineering (Chhabra and Richardson 1999). Another example is in the numerical study of bubbly flows without any contaminants in a clean system, in which it is usually more appropriate to assume that the surface boundary condition of bubbles is free-slip (Kempe et al. 2015). For slip boundary condition, the tangential slip velocity is determined by the surface shear τ w and the wall-normal stress σ w as u| Γ = f (τ w , σ w ) (Rao and Rajagopal 1999). ...
Multiphase flows with momentum, heat, and mass transfer exist widely in a variety of industrial applications. With the rapid development of numerical algorithms and computer capacity, advanced numerical simulation has become a promising tool in investigating multiphase transport problems. Immersed boundary (IB) method has recently emerged as such a popular interface capturing method for efficient simulations of multiphase flows, and significant achievements have been obtained. In this review, we attempt to give an overview of recent progresses on IB method for multiphase transport phenomena. Firstly, the governing equations, the basic ideas, and different boundary conditions for the IB methods are introduced. This is followed by numerical strategies, from which the IB methods are classified into two types, namely the artificial boundary method and the authentic boundary method. Discussions on the implementation of various boundary conditions at the interphase surface with momentum, heat, and mass transfer for different IB methods are then presented, together with a summary. Then, the state-of-the-art applications of IB methods to multiphase flows, including the isothermal flows, the heat transfer flows, and the mass transfer problems are outlined, with particular emphasis on the latter two topics. Finally, the conclusions and future challenges are identified.
